Theorem imaiun1 44650

Description: The image of an indexed union is the indexed union of the images. (Contributed by RP, 29-Jun-2020.)

Assertion

Ref	Expression
imaiun1	⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem imaiun1

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 3290	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
2		vex 3455	. . . . . 6 ⊢ 𝑦 ∈ V
3	2	elima3 6063	. . . . 5 ⊢ (𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
4	3	rexbii 3110	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
5		eliun 4955	. . . . . . 7 ⊢ (⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵)
6	5	anbi2i 635	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵))
7		r19.42v 3195	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵) ↔ (𝑧 ∈ 𝐶 ∧ ∃𝑥 ∈ 𝐴 ⟨𝑧, 𝑦⟩ ∈ 𝐵))
8	6, 7	bitr4i 281	. . . . 5 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
9	8	exbii 1881	. . . 4 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐵))
10	1, 4, 9	3bitr4ri 307	. . 3 ⊢ (∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶))
11	2	elima3 6063	. . 3 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ ∪ 𝑥 ∈ 𝐴 𝐵))
12		eliun 4955	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶) ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 “ 𝐶))
13	10, 11, 12	3bitr4i 306	. 2 ⊢ (𝑦 ∈ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶))
14	13	eqriv 2758	1 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 “ 𝐶) = ∪ 𝑥 ∈ 𝐴 (𝐵 “ 𝐶)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3087  ⟨cop 4590  ∪ ciun 4951   “ cima 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-iun 4953  df-br 5104  df-opab 5168  df-xp 5657  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664

This theorem is used by:  trclimalb2  44725